On weighted extremal K\"ahler metrics

TL;DR

This paper surveys recent developments in weighted extremal Kähler metrics, a generalization of classical extremal metrics that encompasses various important geometric structures.

Contribution

It provides an overview of recent progress related to the YTD conjecture for weighted extremal Kähler metrics.

Findings

Weighted extremal Kähler metrics include Kähler-Ricci solitons and Sasaki-Einstein metrics.

The paper discusses how this notion extends classical extremal metrics.

Recent work on the YTD conjecture for these metrics is summarized.

Abstract

The notion of weighted extremal K\"ahler metrics extends the classical notion of Calabi's extremal K\"ahler metrics, but includes many well-studied objects in K\"ahler geometry such as K\"ahler-Ricci solitons and Sasaki-Einstein metrics. In this paper, after explaining how this notion grew out, we will try to survey recent works concerning the YTD conjecture on weighted extremal K\"ahler metrics.